Finite Group Databasebeta

← #27417064

Subgroups — #27417064

Order 13568.

1364 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810934C211yesmaximal, characteristic, complemented
2C22×#13133141C211yesmaximal, characteristic, complemented
3#20279023C211yesmaximal, characteristic, complemented
4#20275951C212yesmaximal, complemented
5#20275915C212yesmaximal, complemented
6#20275144C211yesmaximal, characteristic, complemented
7#20275008C211yesmaximal, characteristic, complemented
8#20274969C212yesmaximal, complemented
9#20274655C212yesmaximal, complemented
10#20274650C211yesmaximal, characteristic, complemented
11C2×C53×#548C211yesmaximal, characteristic
12C23×#88322C2211yescharacteristic
13C2×#13133198—21no—
14C22×C4⋊Dic53C2211yescharacteristic
15#16810934—24no—
16C2×#13133150C2212yes—
17C2×#13133141—24no—
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133141C2212yes—
22C2×#13133138C2212yes—
23C2×#13133138C2212yes—
24C2×C22⋊#88322C2211yescharacteristic
25C2×C22⋊#88322C2212yes—
26#16810553—21no—
27#16810548—21no—
28#16810521C2212yes—
29#16810484—21no—
30#16810483C2211yescharacteristic
31#16810483C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34C22×#41×C53C2211yescharacteristic
35C23×Q8×C53C2211yescharacteristic
36C53×#548—24no—
37C53×#548—24no—
38C2×#124×C53—21no—
39C2×#124×C53C2211yescharacteristic
40C2×C22⋊Q8×C53—21no—
41C2×C22⋊Q8×C53C2211yescharacteristic
42C2×#121×C53—21no—
43C2×#121×C53C2211yescharacteristic
44C2×C4×Q8×C53C2211yescharacteristic
45C2×C4×D4×C53—21no—
46C2×C4×D4×C53C2211yescharacteristic
47C23×Dic53—21no—
48C22×#88322—22no—
49C22×#88322—42no—
50C22×#88322C2312yes—
51C22×#88322C2312yes—
52#13133198—44no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53—22no—
56C2×C4⋊Dic53C2312yes—
57C2×C4⋊Dic53C2312yes—
58C2×#11719919—22no—
59C2×C22⋊Dic53—22no—
60C2×C22⋊Dic53—21no—
61#13133150—24no—
62#13133141—24no—
63#13133141—24no—
64#13133141—24no—
65#13133141—42no—
66#13133138—24no—
67#13133138—42no—
68C22⋊#88322—22no—
69C22⋊#88322—42no—
70#13133124—22no—
71#13133124—21no—
72#13133124—22no—
73C2×#41×C53—22no—
74C2×#41×C53—22no—
75C2×#41×C53—24no—
76C2×#41×C53—42no—
77C22×Q8×C53—21no—
78C22×Q8×C53—22no—
79C22×Q8×C53—24no—
80C22×Q8×C53—42no—
81C22×Q8×C53D414yes—
82C22×Q8×C53C2311yescharacteristic
83C22×Q8×C53C2311yescharacteristic
84C22×D4×C53—21no—
85C22×D4×C53—21no—
86C22×D4×C53D411yescharacteristic
87C22×D4×C53C2311yescharacteristic
88C23×C4×C53—21no—
89C23×C4×C53—21no—
90C23×C4×C53C2311yescharacteristic
91C23×C4×C53C2311yescharacteristic
92#124×C53—28no—
93#124×C53—22no—
94#124×C53—22no—
95C22⋊Q8×C53—28no—
96C22⋊Q8×C53—22no—
97C22⋊Q8×C53—22no—
98#121×C53—28no—
99#121×C53—22no—
100#121×C53D414yes—
101C4×Q8×C53—22no—
102C4×Q8×C53—22no—
103C4×D4×C53—28no—
104C4×D4×C53—22no—
105C4×D4×C53—22no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
118C24×C53—21no—
119C24×C53C2×D411yescharacteristic
120C24×C53C2×D411yescharacteristic
121#41×C53—48no—
122#41×C53—44no—
123#41×C53—44no—
124C2×Q8×C53—24no—
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—22no—
128C2×Q8×C53—28no—
129C2×Q8×C53—22no—
130C2×Q8×C53—22no—
131C2×Q8×C53—44no—
132C2×Q8×C53—44no—
133C2×Q8×C53—48no—
134C2×Q8×C53C2×D412yes—
135C2×Q8×C53C2×D412yes—
136C2×Q8×C53#4114yes—
137C2×D4×C53—24no—
138C2×D4×C53—22no—
139C2×D4×C53—24no—
140C2×D4×C53—22no—
141C2×D4×C53—22no—
142C2×D4×C53—21no—
143C2×D4×C53—22no—
144C2×D4×C53—22no—
145C2×D4×C53—44no—
146C2×D4×C53—42no—
147C2×D4×C53—42no—
148C2×D4×C53C2×D412yes—
149C22×C4×C53—21no—
150C22×C4×C53—22no—
151C22×C4×C53—22no—
152C22×C4×C53—24no—
153C22×C4×C53—24no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—21no—
158C22×C4×C53—22no—
159C22×C4×C53—22no—
160C22×C4×C53—44no—
161C22×C4×C53—44no—
162C22×C4×C53C2×D411yescharacteristic
163C22×C4×C53C2×D411yescharacteristic
164C22×C4×C53C2×D411yescharacteristic
165C22×C4×C53C2×D411yescharacteristic
166C22×C4×C53C2×D412yes—
167C22×C4×C53C2×D412yes—
168C22×C4×C53C2411yescharacteristic
169C22×Dic53—22no—
170C22×Dic53—22no—
171C22×Dic53—42no—
172C22×Dic53—42no—
173C22×Dic53—42no—
174C4⋊C4×C53—24no—
175C4⋊C4×C53—24no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53C2×D412yes—
179C22⋊C4×C53—24no—
180C22⋊C4×C53—24no—
181C22⋊C4×C53—24no—
182C22⋊C4×C53—22no—
183C22⋊C4×C53—24no—
184C22⋊C4×C53—22no—
185C42×C53—24no—
186C42×C53—21no—
187C42×C53—21no—
188C22⋊Dic53—44no—
189C22⋊Dic53—42no—
190C4⋊Dic53—22no—
191C4⋊Dic53—22no—
192C4⋊Dic53—22no—
193C4⋊Dic53—22no—
194C4⋊Dic53—42no—
195#11719919—44no—
196C2×#88322—22no—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322—22no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—44no—
203C23×C53—22no—
204C23×C53—22no—
205C23×C53—21no—
206C23×C53—22no—
207C23×C53—22no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—22no—
211C23×C53—42no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—42no—
215C23×C53#12012yes—
216C23×C53#13512yes—
217C23×C53AGL(1,ℤ/8)12yes—
218C23×C53C22×D411yescharacteristic
219Q8×C53—22no—
220Q8×C53—24no—
221Q8×C53—22no—
222Q8×C53—22no—
223Q8×C53—48no—
224Q8×C53—42no—
225Q8×C53#12314yescomplemented
226D4×C53—24no—
227D4×C53—48no—
228D4×C53—44no—
229D4×C53—48no—
230D4×C53—41no—
231D4×C53—41no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—24no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—24no—
238C2×C4×C53—21no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—22no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—22no—
247C2×C4×C53—48no—
248C2×C4×C53—48no—
249C2×C4×C53—42no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—42no—
255C2×C4×C53#12012yes—
256C2×C4×C53C22×D411yescharacteristic
257C2×C4×C53C22×D411yescharacteristic
258C2×C4×C53C2×#4112yes—
259C2×C4×C53H(5,𝔽2)12yes—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—84no—
267#88322—42no—
268#88322—42no—
269#24035—531nomaximal, Sylow
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—24no—
273C22×C53—24no—
274C22×C53—42no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—42no—
278C22×C53—41no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—82no—
283C22×C53—82no—
284C22×C53#44914yes—
285C22×C53C2×#12011yescharacteristic
286C22×C53#54612yes—
287C22×C53C2×#13511yescharacteristic
288C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C53#57812yes—
290C4×C53—22no—
291C4×C53—24no—
292C4×C53—41no—
293C4×C53—42no—
294C4×C53—41no—
295C4×C53—42no—
296C4×C53—41no—
297C4×C53C2×#12311yescharacteristic
298C4×C53#53411yescharacteristic
299C4×C53#54612yes—
300Dic53—82no—
301Dic53—82no—
302C2×#548D5311yescharacteristic
303C2×#480—531no—
304C22×#103—531no—
305#2486—531no—
306#2042—532no—
307#2036—532no—
308#1893—531no—
309#1869—531no—
310#1863—532no—
311#1788—532no—
312#1787—531no—
313C2×C53—24no—
314C2×C53—81no—
315C2×C53—82no—
316C2×C53#240511yescharacteristic
317C2×C53C2×#44911yescharacteristic
318C2×C53#301111yescharacteristic
319C2×C53#330012yes—
320C2×C53#330812yes—
321C22×#41C2×D5311yescharacteristic
322C23×Q8C2×D5311yescharacteristic
323C23×C8—531no—
324#548—24no—
325#548—24no—
326C2×#124—21no—
327C2×#124C2×D5311yescharacteristic
328C2×#123—1061no—
329C2×C22⋊Q8—21no—
330C2×C22⋊Q8C2×D5311yescharacteristic
331C2×#121—21no—
332C2×#121C2×D5311yescharacteristic
333C2×C4×Q8C2×D5311yescharacteristic
334C2×C4×D4—21no—
335C2×C4×D4C2×D5311yescharacteristic
336C22×C4⋊C4—531no—
337#480—1064no—
338C2×#107—532no—
339C2×#103—532no—
340C2×#103—532no—
341C2×#103—532no—
342C2×#103—532no—
343C2×#103—1064no—
344C2×#102—532no—
345C2×#102—532no—
346C2×C22⋊C8—531no—
347C2×C22⋊C8—532no—
348#398—1061no—
349#397—1061no—
350#389—532no—
351#381—1061no—
352#380—531no—
353#380—532no—
354#340—532no—
355#336—532no—
356C53#2403511yescharacteristic, complemented, Sylow
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—24no—
360C2×#41—42no—
361C22×Q8—21no—
362C22×Q8—22no—
363C22×Q8—24no—
364C22×Q8—42no—
365C22×Q8C22⋊D5314yes—
366C22×Q8C22×D5311yescharacteristic
367C22×Q8C22×D5311yescharacteristic
368C22×D4—21no—
369C22×D4—21no—
370C22×D4C22⋊D5311yescharacteristic
371C22×D4C22×D5311yescharacteristic
372C23×C4—21no—
373C23×C4—21no—
374C23×C4—1061no—
375C23×C4C22×D5311yescharacteristic
376C23×C4C22×D5311yescharacteristic
377C22×C8—532no—
378C22×C8—532no—
379C22×C8—1062no—
380C22×C8—2122no—
381#124—28no—
382#124—22no—
383#124—22no—
384#123—2124no—
385C22⋊Q8—28no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388#121—28no—
389#121—22no—
390#121C22⋊D5314yes—
391C4×Q8—22no—
392C4×Q8—22no—
393C4×D4—28no—
394C4×D4—22no—
395C4×D4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—532no—
399C2×C4⋊C4—532no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1062no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1062no—
404C2×C4⋊C4C22⋊D5311yescharacteristic
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1062no—
412C2×C22⋊C4C22⋊D5311yescharacteristic
413C2×C22⋊C4C22⋊D5311yescharacteristic
414C2×C42—21no—
415C2×C42C22×D5311yescharacteristic
416#107—1064no—
417#103—1064no—
418#103—1064no—
419#103—1064no—
420#103—2122no—
421#102—1064no—
422#102—2122no—
423C22⋊C8—1062no—
424C22⋊C8—2122no—
425#95—1062no—
426#95—1061no—
427#95—1062no—
428C24—21no—
429C24C2×C22⋊D5311yescharacteristic
430C24C2×C22⋊D5311yescharacteristic
431#41—48no—
432#41—44no—
433#41—44no—
434C2×Q8—24no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—28no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—48no—
444C2×Q8#1171994714yes—
445C2×Q8C2×C22⋊D5312yes—
446C2×Q8C2×C22⋊D5312yes—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4C2×C22⋊D5312yes—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—2122no—
475C22×C4—2122no—
476C22×C4—2122no—
477C22×C4C2×C22⋊D5311yescharacteristic
478C22×C4C2×C22⋊D5311yescharacteristic
479C22×C4C2×C22⋊D5311yescharacteristic
480C22×C4C2×C22⋊D5311yescharacteristic
481C22×C4C2×C22⋊D5312yes—
482C22×C4C2×C22⋊D5312yes—
483C22×C4C23×D5311yescharacteristic
484C2×C8—1062no—
485C2×C8—1062no—
486C2×C8—1062no—
487C2×C8—1062no—
488C2×C8—1062no—
489C2×C8—1062no—
490C2×C8—2124no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—1062no—
496C4⋊C4—1062no—
497C4⋊C4—1062no—
498C4⋊C4—1062no—
499C4⋊C4—2124no—
500C4⋊C4—2122no—
501C4⋊C4C2×C22⋊D5312yes—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—24no—
507C22⋊C4—22no—
508C22⋊C4—2124no—
509C22⋊C4—2122no—
510C42—24no—
511C42—21no—
512C42—21no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—21no—
517C23—22no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23#1313318612yes—
526C23#1313323712yes—
527C23#1313324212yes—
528C23C22×C22⋊D5311yescharacteristic
529Q8—24no—
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8—48no—
534Q8—42no—
535Q8#1313319814yescomplemented
536D4—24no—
537D4—48no—
538D4—44no—
539D4—48no—
540D4—41no—
541D4—41no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—4244no—
566C2×C4—2122no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4—2122no—
571C2×C4—2122no—
572C2×C4#1313318612yes—
573C2×C4C22×C22⋊D5311yescharacteristic
574C2×C4C22×C22⋊D5311yescharacteristic
575C2×C4C2×#1171994712yes—
576C2×C4#1313326212yes—
577C8—2122no—
578C8—2122no—
579C22—22no—
580C22—22no—
581C22—24no—
582C22—24no—
583C22—41no—
584C22—42no—
585C22—42no—
586C22—41no—
587C22—41no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—82no—
592C22—82no—
593C22#1681076814yes—
594C22C2×#1313318611yescharacteristic
595C22#1681126712yes—
596C22C2×#1313323711yescharacteristic
597C22C2×#1313324211yescharacteristic
598C22#1681143812yes—
599C4—22no—
600C4—24no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—4242no—
607C4—4242no—
608C4C2×#1313319811yescharacteristic
609C4#1681118211yescharacteristic
610C4#1681126712yes—
611C2—24no—
612C2—81no—
613C2—82no—
614C2#2027803811yescharacteristic
615C2C2×#1681076811yescharacteristic
616C2#2028401011yescharacteristic
617C2#2028660212yes—
618C2#2028670612yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#548C2
socleC23×C53C22×D4
hypercenterC2×#548D53
hypocenterC53#24035
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#548

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16810934C2semidirect
2. C22×#13133141C2semidirect
3. #20279023C2semidirect
4. #20275951C2semidirect
5. #20275915C2semidirect
6. #20275144C2semidirect
7. #20275008C2semidirect
8. #20274969C2semidirect
9. #20274655C2semidirect
10. #20274650C2semidirect
225. Q8×C53#123semidirect
356. C53#24035semidirect
535. Q8#13133198semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168109342
C22×#131331412
#202790232
#202759512
#202759152
#202751442
#202750082
#202749692
#202746552
#202746502
C2×C53×#5482
#2403553